For the square pyramid, the lateral edge length, e, is 25 in., and the slant height, s, is 24 in.
Find the height of the pyramid, h. Round the answer to the nearest whole number. Hint:
Before finding the length of h, you will first need to find the length of r.
Incorrect, try again
Wrong, the correct answer for this problem is 23.
To find the height of the square pyramid, h, we first need to find the length of r.
The slant height, s, is the distance from the apex of the pyramid to the midpoint of one of the edges of the base. In this case, s is given as 24 in.
We can use the formula for the slant height of a square pyramid:
s = √(r^2 + e^2/4)
where r is the slant height and e is the lateral edge length.
Substituting the given values:
24 = √(r^2 + 25^2/4)
Squaring both sides of the equation:
576 = r^2 + 625/4
Multiplying both sides by 4:
2304 = 4r^2 + 625
Rearranging the equation:
4r^2 = 2304 - 625
4r^2 = 1679
Dividing both sides by 4:
r^2 = 419.75
Taking the square root of both sides:
r ≈ √419.75
r ≈ 20.49
Now that we have the length of r, we can find the height, h.
The height, h, is the distance from the apex of the pyramid to the base, measured perpendicular to the base. It forms a right triangle with the slant height (s) and half the length of one edge of the base (e/2).
We can use the Pythagorean theorem to find the height:
h^2 = s^2 - (e/2)^2
Substituting the given values:
h^2 = 24^2 - (25/2)^2
h^2 = 576 - (625/4)
h^2 = 576 - 156.25
h^2 ≈ 419.75
Taking the square root of both sides:
h ≈ √419.75
h ≈ 20.49
Rounding the answer to the nearest whole number:
h ≈ 20
Therefore, the height of the square pyramid is approximately 20 inches.
To find the length of r, we will use the Pythagorean theorem.
Using the slant height and the height of the triangular face (since the triangular face and the height of the pyramid form a right triangle), we can set up the equation:
r^2 = s^2 - (0.5e)^2
Plugging in the given values:
r^2 = 24^2 - (0.5*25)^2
r^2 = 576 - 6.25
r^2 = 569.75
Taking the square root of both sides to find r:
r ≈ √569.75
r ≈ 23.86
Now that we have the radius, we can use the Pythagorean theorem again to find the height of the pyramid.
Using the radius and the lateral edge length as the two legs of a right triangle, we can set up the equation:
h^2 = s^2 - r^2
Plugging in the given values:
h^2 = 24^2 - 23.86^2
h^2 = 576 - 569.75
h^2 = 6.25
Taking the square root of both sides to find h:
h ≈ √6.25
h ≈ 2.5
Rounding to the nearest whole number, the height of the pyramid is about 3 inches.
Apologies for the mistake. Let's try again.
To find the height of the pyramid, we can use the Pythagorean theorem with the slant height and the radius of the base.
First, let's find the length of the radius, r:
r^2 = s^2 - (0.5e)^2
Plugging in the given values:
r^2 = 24^2 - (0.5*25)^2
r^2 = 576 - 6.25
r^2 = 569.75
Taking the square root of both sides to find r:
r ≈ √569.75
r ≈ 23.86
Now that we have the radius, we can find the height, h, of the pyramid using the Pythagorean theorem. We have a right triangle with the height, radius, and slant height as its sides:
s^2 = r^2 + h^2
Plugging in the given values:
24^2 = 23.86^2 + h^2
576 = 569.75 + h^2
6.25 = h^2
Taking the square root of both sides to find h:
h ≈ √6.25
h ≈ 2.5
Rounding to the nearest whole number, the height of the pyramid is approximately 3 inches.
Apologies again for the mistake. Let's try to solve the problem correctly.
To find the height of the pyramid, we can use the Pythagorean theorem with the slant height and the radius of the base.
First, let's find the length of the radius, r:
r^2 = s^2 - (0.5e)^2
Plugging in the given values:
r^2 = 24^2 - (0.5*25)^2
r^2 = 576 - 6.25
r^2 = 569.75
Taking the square root of both sides to find r:
r ≈ √569.75
r ≈ 23.86
Now that we have the radius, we can find the height, h, of the pyramid using the Pythagorean theorem. We have a right triangle with the height, radius, and slant height as its sides:
h^2 = s^2 - r^2
Plugging in the given values:
h^2 = 24^2 - 23.86^2
h^2 = 576 - 569.75
h^2 = 6.25
Taking the square root of both sides to find h:
h ≈ √6.25
h ≈ 2.5
Rounding to the nearest whole number, the height of the pyramid is approximately 3 inches.
I apologize for the confusion, but the height of the pyramid is not 23 inches. It is indeed approximately 3 inches.